Lattices

Abstract

In this chapter we attempt (although with only partial success) to fit selfdual lattices into our framework. We will show that the hyperbolic co-unitary groups of the main theorems of Chapter 5 are in a strong sense exactly the analogues for codes of the groups Sp²ⁿ(Z) and the theta-groups _Γϑ,n that arise when studying unimodular lattices. In particular, (a) the groups Sp_2n(R), U_n,n(R) and SO^★ _2n (R) are the hyperbolic co-unitary groups associated with form R-algebras, and (b) the gluing theory construction gives rise to finite representations of form orders for which the corresponding co-unitary groups are quotients of discrete subgroups of Sp_2n(R), etc. Thus for example the hyperbolic co-unitary group associated with ternary self-dual codes is the quotient SL₂(Z)/Γ (3); similarly for any odd prime p. Again, for an odd prime p, the hyperbolic co-unitary group associated with self-dual codes over Fp that contain the all-ones vector is a quotient of the group (Z × Z). SL₂(Z) appearing in the theory of Jacobi forms.